# The correct option among the following is?

Feb 27, 2018

#### Explanation:

We can write $9 \left(25 {a}^{2} + {b}^{2}\right) + 25 \left({c}^{2} - 3 a c\right) = 15 b \left(3 a + c\right)$ as

$225 {a}^{2} + 9 {b}^{2} + 25 {c}^{2} - 75 a c = 45 a b + 15 b c$

or $225 {a}^{2} + 9 {b}^{2} + 25 {c}^{2} - 75 a c - 45 a b - 15 b c = 0$

or ${\left(15 a\right)}^{2} + {\left(3 b\right)}^{2} + {\left(5 c\right)}^{2} _ \left(15 a\right) \left(5 c\right) - \left(15 a\right) \left(3 b\right) - \left(3 b\right) \left(5 c\right) = 0$

This is of type ${u}^{2} + {v}^{2} + {w}^{2} - u v - v w - u w = 0$

and we can write it as $2 {u}^{2} + 2 {v}^{2} + 2 {w}^{2} - 2 u v - 2 v w - 2 u w = 0$

or ${\left(u - v\right)}^{2} + {\left(v - w\right)}^{2} + {\left(u - w\right)}^{2} = 0$

As sum of three squares is $0$, each one of them is $0$

and hence $u = v = w$ and in given case we have

$15 a = 3 b = 5 c$ and $b = 5 a$ and $c = 3 a$

Hence our $a , b$ and $c$ are equivalent to $a , 5 a$ and $3 a$

and $a , c$ and $b$ - or $b , c$ and $a$ are in A.P.